Arthmatic and Ceametric Senuences Review Arimmetic bequence is a sequance of temis that have a common) $$ \qquad $$ ? bywien them Equation Used: $$ a_{n}=a_{1}+d(n-1) $$ $$ \begin{array}{l} e_{i}= \ n= \ B= \end{array} $$ Geometric Sequence - is a sequence of terms that have a common $$ \qquad $$ between them. Equation Used: $$ a_{n}=a_{1} r^{(n-1)} $$ $$ \begin{array}{l} a_{1}= \ {\left[\begin{array}{l} = \ = \ b= \end{array}\right.} \end{array} $$ Determine if the following sequences are anthmetic. If it is, what is the common difference? 1. $$ -9,-109,-209,-309 $$... $$ 2.28,18,8,2 $$ $$ 3,28,26,24,22, \ldots $$ 4. $$ -16,-6,4,14 $$. Find the common difference and the next three terms in the sequence 5. $$ 39,-33,-27-21, \ldots $$ 6. $$ -27,-17,-7,3 $$

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Let's break down the problem step by step.

### Definitions
1. **Arithmetic Sequence**: A sequence of terms that have a common **difference** between them. The equation used is:
   $$
   a_{n} = a_{1} + d(n-1)
   $$
   where:
   - $$ e_{i} $$ is the $$ i $$-th term,
   - $$ n $$ is the term number,
   - $$ d $$ is the common difference.

2. **Geometric Sequence**: A sequence of terms that have a common **ratio** between them. The equation used is:
   $$
   a_{n} = a_{1} r^{(n-1)}
   $$
   where:
   - $$ a_{1} $$ is the first term,
   - $$ r $$ is the common ratio.

### Determine if the following sequences are arithmetic and find the common difference:

1. **Sequence**: $$ -9, -109, -209, -309 $$
   - Common difference $$ d = -109 - (-9) = -100 $$
   - Yes, it is arithmetic.

2. **Sequence**: $$ 2, 28, 18, 8, 2 $$
   - Common differences:
     - $$ 28 - 2 = 26 $$
     - $$ 18 - 28 = -10 $$
     - $$ 8 - 18 = -10 $$
     - $$ 2 - 8 = -6 $$
   - No, it is not arithmetic.

3. **Sequence**: $$ 3, 28, 26, 24, 22, \ldots $$
   - Common differences:
     - $$ 28 - 3 = 25 $$
     - $$ 26 - 28 = -2 $$
     - $$ 24 - 26 = -2 $$
     - $$ 22 - 24 = -2 $$
   - No, it is not arithmetic.

4. **Sequence**: $$ -16, -6, 4, 14 $$
   - Common difference $$ d = -6 - (-16) = 10 $$
   - Yes, it is arithmetic.

### Find the common difference and the next three terms in the sequence:

5. **Sequence**: $$ 39, -33, -27, -21, \ldots $$
   - Common difference $$ d = -33 - 39 = -72 $$
   - Next three terms:
     - $$ -21 - 72 = -93 $$
     - $$ -93 - 72 = -165 $$
     - $$ -165 - 72 = -237 $$
   - Next three terms: $$ -93, -165, -237 $$

6. **Sequence**: $$ -27, -17, -7, 3 $$
   - Common difference $$ d = -17 - (-27) = 10 $$
   - Next three terms:
     - $$ 3 + 10 = 13 $$
     - $$ 13 + 10 = 23 $$
     - $$ 23 + 10 = 33 $$
   - Next three terms: $$ 13, 23, 33 $$

### Summary of Results
1. Arithmetic, common difference: $$ -100 $$
2. Not arithmetic
3. Not arithmetic
4. Arithmetic, common difference: $$ 10 $$
5. Common difference: $$ -72 $$, next three terms: $$ -93, -165, -237 $$
6. Common difference: $$ 10 $$, next three terms: $$ 13, 23, 33 $$
**Arithmetic and Geometric Sequences Review** 1. **Arithmetic Sequence**: A sequence where each term increases or decreases by a common **difference**. The formula is $$ a_{n} = a_{1} + d(n-1) $$. 2. **Geometric Sequence**: A sequence where each term is multiplied by a common **ratio**. The formula is $$ a_{n} = a_{1} r^{(n-1)} $$. **Determining if Sequences are Arithmetic and Finding Common Differences:** 1. **Sequence**: $$ -9, -109, -209, -309 $$ - **Common Difference**: $$ -100 $$ - **Result**: Arithmetic 2. **Sequence**: $$ 2, 28, 18, 8, 2 $$ - **Common Differences**: $$ 26, -10, -6 $$ - **Result**: Not Arithmetic 3. **Sequence**: $$ 3, 28, 26, 24, 22, \ldots $$ - **Common Differences**: $$ 25, -2, -2, -2 $$ - **Result**: Not Arithmetic 4. **Sequence**: $$ -16, -6, 4, 14 $$ - **Common Difference**: $$ 10 $$ - **Result**: Arithmetic **Finding Common Differences and Next Three Terms:** 5. **Sequence**: $$ 39, -33, -27, -21, \ldots $$ - **Common Difference**: $$ -72 $$ - **Next Three Terms**: $$ -93, -165, -237 $$ 6. **Sequence**: $$ -27, -17, -7, 3 $$ - **Common Difference**: $$ 10 $$ - **Next Three Terms**: $$ 13, 23, 33 $$ **Summary:** - Sequences 1 and 4 are arithmetic with common differences of $$ -100 $$ and $$ 10 $$ respectively. - Sequences 2 and 3 are not arithmetic. - For sequences 5 and 6, the common differences are $$ -72 $$ and $$ 10 $$, and the next three terms are $$ -93, -165, -237 $$ and $$ 13, 23, 33 $$ respectively.

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Definitions

Determine if the following sequences are arithmetic and find the common difference:

Find the common difference and the next three terms in the sequence:

Summary of Results

The Deep Dive

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